f*- Coercive function
Alyaa Yousif Khudayir Habeeb Kareem Abdullah
University of kufa, College of Education for Girls, Department of Mathematics
Abstract
In this paper , we introduce the definition of f* - coercive function and introduce several properties of f* - coercive function .
الخلاصة
في هذا البحث، قدمنا تعريف لـ f* - coercive function وقدمنا بعض مبرهنات لـ
f* - coercive function
Introduction
Let be a topological space. Asubset A of a space X is called semi- open if and A is called feebly open ( f- open) if there exists an open set U of X such that where stands for the intersection of all semi-closed subsets of X which contain U ,(Navalagi, 1991).
In this paper gives anew definition namely of f* - coercive function .
1-Basic concepts
Definition 1.1,(Levine, 1963).
A set B in a space X is called semi – open (s.o) if there exists an open subset O of X such that .
The complement of a semi – open set is defined to be semi – closed (s.c.)
Definition 1.2,(Dorsett, 1981).
Let X be a space and . Then the intersection of all semi – closed subsets of X which contains A is called semi – closure of A and it is denoted by .
Definition 1.3,(Dontchev, 1998).
A subset B of a space X is called pre – open if . The complement of a pre –open set is defined to be pre – closed .
Definition 1.4,(Navalagi, 1991).
A subset B of a space X is called feebly open (f-open) set if there exists open subset U of X such that .
The complement of a feebly open set is defined to be a feebly closed
(f-closed)set .
Proposition1.5(Farero, 1987).
Let X be a space and . Then the following statements are equivalent :
( i ) B is f – open set .
( ii ) There exists an open set O in X such that .
( iii ) B is semi – open and pre – open .
Definition 1.6(Maheshwari, 1985)
A space X is called f-compact if every f-open cover of X has a finite subcover.
Lemma 1.7(Khudayir, 2008)
Let X be space and F be an f-closed subset of X, then is
f- compact subset of F, for every f-compact set K in X .
Definition 1.8, (Khudayir, 2008)
Let X and Y be spaces, the function is called st-f-compact if the inverse image of each f-compact set in Y is f- compact set in X.
Definition 1.9 (Khudayir, 2008)
Let X and Y be spaces .A function is called f- coercive if for every f -compact set , there exists f- compact set such that :
Definition 1.10,(Maheshwari and Thakur, 1980; Reilly and Vammanamurthy, 1985; Navalagi, 1998)
Let X and Y be spaces and be a function,Then f is called
f-continuous function if is an f- open set in X for every open set A in Y.
Definition 1.11,( Reilly and Vammanamurthy, 1985; Navalagi, 1998)
A function is called st-f-closed function if the image of each f- closed subset of X is an f-closed set in Y .
Definition 1.12,(Khudayir, 2008)
Let X and Y be spaces .Then is called a strong feebly proper (st-f-proper) function if :
(i)f is f-continuous function.
(ii) : is ast-f-closed function , for every space Z.
Proposition 1.13, (Khudayir, 2008)
Let be a function on a space X. If f is st-f-proper, then X is an f-compact space, where w is any point which dose not belong to X .
2- The main results
Definition (2.1) :
Let X and Y be spaces .A function is called
f* - coercive if forevery f – compact set , there exists compact set such that :
Example (2.2):
If X is compact space, then the function is f* - coercive .
Remark 2.3 :
Every f - coercive function is f* - coercive function .
.
Proposition (2.4):
Let be st-f-proper function ,then is f* - coercive function ; where w is any point which dose not belongto X .
Proof :
By proposition (1.13) and Example(2.2) .
Proposition (2.5):
For any f- closed subset F of a space X , the inclusion function is f* - coercive function .
Proof:
Let J be an f-compact subset of X, then by lemma (1.7), is f-compact set in F , then is compact set in F .
But
Therefore the inclusion function is f* - coercive function .
Proposition (2.6):
If is st-f-compact function, then is
f* - coercive function .
Proof:
Let J be an f-compact set in Y, since is st-f-compact function, then is f- compact set in X , thus is compact set in X .
Thus
Thereforeis f* - coercive function .
Proposition (2.7):
Let X ,Y and Z be spaces. If is f*-coercive andis f-coercive function,then gof is a f*-coercive function
.
Proof:
Let J be an f-compact set in Z ,then there exists f-compact set K in Y such that:
Since is f* - coercive function, then there exists a compact set D in X such that
Then
Therefore is f* - coercive function .
Proposition (2.8):
Let be a f* - coercive function such that F is f- closed subset of X. Then is af* - coercive function .
Proof:
Since F is f- closed set in Y , then by proposition(2.5), the inclusion function is f* - coercive function, since is
f* - coercive function, then by proposition (2.7). is
f* - coercive function.
But , then is f* - coercive function.
Proposition (2.9):
Let X and Y be spaces, such that Y is and
is continuous , one – one , function .Then the following statements are equivalent :
(i) f is an f*-coercive function .
(ii)f is an f- compact function .
(iii)f is an f- proper function .
Proof:
(i → ii) Let J be an f-compact set in Y. To prove is acompact set in X . Let be a net in . Since f is f*-coercive function, then there exists a compact set K in X such that then
Then . Then is a net in K , Since K is a compact set in X , then by [Reilly and Vammanamurthy, 1985,theorem 3.15],the net has a cluster point x in X .Thus by [Reilly and Vammanamurthy, 1985,theorem 3.15], is acompact set in X.
Therefore is an f-compact function .
(ii → iii ) By [AL-Badairy,2005 , proposition 3.1.22]
( iii → i ) let J be an f-compact set in Y , since f is f- proper function , then by [AL-Badairy,2005,proposition 3.1.21], f is f- compact function, then is a compact in X . Thus
Hence is f*- coercive function .
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